Solving Harmonic Oscillator Coefficients w/ Hermite Polynomials

[Sojourner01]

Ok, fairly basic quantum mechanics assignment.

One question deals with (I think) the coefficients of the Hermite polynomial. Unfortunately, the lecturer hasn't told us anything about this method, so I donn't know what it's called or what the point of it is, and it's not in any of the examples of Hermite polynomials I can find.

I have the summation:

$$\\sum_{n=-\\infty}^\\infty [(k+2)(k+1)c_{k+2} + (2 \\epsilon -2k -1)c_{k}] y^k = 0$$

Show that the above implies that the coefficients for each power of y are themselves zero, by considering the derivatives of [] evaluated at y=0?

It'd be nice if I knew what these damn numbers were. It'd be even nicer if I knew what this was called so I could look it up.

edit: well, I can't get the LaTex to display what I want, but I hope you get the idea...

 

[Galileo]

Use a single backslash before LaTeX command:
$$\sum_{n=-\infty}^\infty [(k+2)(k+1)c_{k+2} + (2 \epsilon -2k -1)c_{k}] y^k = 0$$